Question

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question.

Caffeine in Soft Drinks Listed below are measured amounts of caffeine (mg per 12 oz of drink) obtained in one can from each of 20 brands (7-UP, A&W Root Beer, Cherry Coke, . . . , Tab). Are the statistics representative of the population of all cans of the same 20 brands consumed by Americans? 0 0 34 34 34 45 41 51 55 36 47 41 0 0 53 54 38 0 41 47

Short answer

(a) The mean is 32.6 mg.

(b) The median is 39.5 mg

(c) The mode is 0 mg.

(d) The midrange is 27.5 mg.

The resulting statistics are not considered to be the representative of the population.

Step-by-step solution

Given information

Twenty brands of soft drinks resulted in the following caffeine amounts:

0
0
34
34
34
45
41
51
55
36
47
41
0
0
53
54
38
0
41
47

Compute mean

(a)

Mean is computed using the formula given below.

$\overline{x}=\frac{\sum x}{n}$, where xrepresents the observations and nis the count of the observations.

Substitute the values in the formula.

$$\begin{gathered} \overline{x}=\frac{0+0+...+47}{20} \\ =\frac{651}{20} \\ =32.55 \\ \approx 32.6 \end{gathered}$$

Thus, the mean value is approximately 32.6 mg.

Compute median

(b)

Median is computed using the following criteria:

• Median is equal to the middlemost observation when n is an odd value.
• Median is equal to the average of the two middle observations when n is an even value.

The number of observations is 20.

Arrange the observations in ascending order.

0
0
0
0
0
34
34
34
36
38
41
41
41
45
47
47
51
53
54
55

The middlemost observations are 38 and 41.

The median is given as:

$$\begin{gathered} M=\frac{38+41}{2} \\ =\frac{79}{2} \\ =39.5 \end{gathered}$$

Thus, the median is 39.5 mg.

Compute mode

(c)

The observation with the maximum frequency is the mode.

The frequency distributionsfor the amount of caffeine are:

Amount of caffeine	Frequency
0	5
34	3
36	1
38	1
41	3
45	1
47	2
51	1
53	1
54	1
55	1

The highest frequency is obtained for 0mg.

Thus, the mode of the data is 0 mg.

Compute midrange

(d)

The midrange is computed using the following formula:

$\text{Midrange}=\frac{\text{Minimum}\text{value}+\text{Maximum}\text{value}}{2}$

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{0+55}{2} \\ =27.5 \end{gathered}$$

Thus, the midrange is 27.5 mg.

Discuss if the statistics are representative of the population

The statistics are not representative of the population because of the following:

• The brands evaluated in the study are not stated, and there are high chances that few brands are much more popular than the others.
• The quantity of caffeine in each of the cans from various brands may differ.

Inferring from the above points, the statistics are not a true representative of the population.